Simpson's Rule Calculator

Approximate definite integrals with Simpson's 1/3 rule, 3/8 rule and the composite rule. Includes an interactive parabolic visualization, error estimation, convergence analysis, method comparison and step-by-step solutions.

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About Simpson's Rule Calculator

The Simpson's Rule Calculator is a powerful numerical integration tool that approximates definite integrals by fitting parabolic curves (1/3 rule) or cubic curves (3/8 rule) through sample points. Unlike the trapezoidal rule that uses straight lines between points, Simpson's rule captures the curvature of the function, delivering O(h⁴) accuracy — making it one of the most widely used methods in calculus, engineering, and scientific computing.

Key Features

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Dual Rule Support

Choose between Simpson's 1/3 rule (quadratic interpolation, even n) and 3/8 rule (cubic interpolation, n divisible by 3). Both are computed with full MathJax step-by-step solutions.

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Interactive Visualization

See parabolic segments drawn under the curve. Hover to highlight individual segments and view their areas. Animate the convergence or use the slider to explore different n values.

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Method Comparison

Automatically compare Simpson's 1/3, 3/8, trapezoidal, and midpoint results side-by-side for the same function and bounds. See which method gives the best approximation.

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Error Analysis

Automatic error bound calculation using the fourth derivative. The error formula $ |E_S| \leq \frac{(b-a)^5}{180n^4} \max|f^{(4)}| $ tells you exactly how accurate your approximation is.

How to Use the Simpson's Rule Calculator

Enter your function — Type a mathematical expression f(x) such as x^2, sin(x), exp(-x^2), or any combination of supported functions.
Set integration bounds — Enter the lower bound (a) and upper bound (b), and choose the number of subintervals (n).
Choose a rule — Select Simpson's 1/3 Rule (requires even n, auto-adjusted if odd) or 3/8 Rule (requires n divisible by 3, auto-adjusted).
Click Calculate — The tool computes the approximation with a complete step-by-step solution rendered in MathJax.
Explore results — Interact with the parabolic visualization, review per-segment areas, compare methods, and study the convergence analysis.

Simpson's 1/3 Rule Explained

The composite Simpson's 1/3 rule divides [a, b] into n equal subintervals (n must be even) and fits a parabola through every three consecutive points:

$$S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \cdots + 4f(x_{n-1}) + f(x_n) \right]$$

where $ \Delta x = \frac{b - a}{n} $. The coefficients follow the pattern 1, 4, 2, 4, 2, ..., 4, 1. Each pair of subintervals uses a quadratic polynomial that passes through three points, capturing the curvature of the function far better than linear interpolation.

Simpson's 3/8 Rule Explained

The 3/8 rule uses cubic interpolation over groups of three subintervals (n must be divisible by 3):

$$S_{3/8} = \frac{3\Delta x}{8} \left[ f(x_0) + 3f(x_1) + 3f(x_2) + 2f(x_3) + 3f(x_4) + \cdots + f(x_n) \right]$$

The coefficients follow the pattern 1, 3, 3, 2, 3, 3, 2, ..., 3, 3, 1. While both rules achieve O(h⁴) accuracy, the 3/8 rule is useful when n is not even.

Error Comparison

Method	Error Order	Error Bound	Exact For
Trapezoidal	$ O(h^2) $	$ \frac{(b-a)^3}{12n^2} \max\|f''\| $	Linear functions
Simpson's 1/3	$ O(h^4) $	$ \frac{(b-a)^5}{180n^4} \max\|f^{(4)}\| $	Cubics and below
Simpson's 3/8	$ O(h^4) $	$ \frac{(b-a)^5}{80n^4} \max\|f^{(4)}\| $	Cubics and below

Doubling n reduces Simpson's rule error by approximately 16×, compared to only 4× for the trapezoidal rule. This makes Simpson's rule converge much faster for smooth functions.

When to Use Each Rule

Simpson's 1/3 rule — Best for most applications. Use when n is even (or can be made even). Most accurate per function evaluation among the three basic Newton-Cotes formulas.
Simpson's 3/8 rule — Use when n is a multiple of 3 but not even. Also useful in composite formulas when combining with the 1/3 rule to handle odd subinterval counts.
Trapezoidal rule — Prefer when data is unevenly spaced, n is odd and small, or simplicity matters more than accuracy. Also better for functions with discontinuities in higher derivatives.

Supported Functions

This calculator supports a wide range of mathematical functions:

Polynomials: x^2, x^3 + 2x - 1, x^5 - 3x^3 + 2
Trigonometric: sin(x), cos(x), tan(x), asin(x), acos(x)
Exponential/Logarithmic: exp(x), ln(x), log(x)
Roots: sqrt(x)
Constants: pi, e
Combinations: sin(x)*exp(-x), x^2/(1+x^2), sqrt(1+x^3)

Frequently Asked Questions

What is Simpson's rule?

Simpson's rule is a numerical integration method that approximates definite integrals by fitting parabolas (or cubic curves) through sample points. It is significantly more accurate than the trapezoidal rule for smooth functions, with an error of order O(h⁴) compared to O(h²).

What is the difference between Simpson's 1/3 and 3/8 rules?

Simpson's 1/3 rule uses quadratic (parabolic) interpolation over pairs of subintervals and requires an even number of subintervals. Simpson's 3/8 rule uses cubic interpolation over triples of subintervals and requires n divisible by 3. The 1/3 rule is generally more accurate for the same n, but the 3/8 rule is useful when n is not even.

Why does Simpson's 1/3 rule require an even number of subintervals?

Simpson's 1/3 rule works by fitting a parabola through three consecutive points (two subintervals at a time). Since each parabolic segment spans exactly 2 subintervals, the total number of subintervals must be even for complete coverage.

How accurate is Simpson's rule compared to the trapezoidal rule?

Simpson's 1/3 rule has an error of O(h⁴), while the trapezoidal rule has O(h²). This means doubling the number of subintervals reduces Simpson's error by about 16 times, compared to only 4 times for the trapezoidal rule. For smooth functions, Simpson's rule is significantly more accurate.

What functions can I enter in the calculator?

You can enter polynomials like x^2 or x^3-2x+1, trigonometric functions like sin(x) and cos(x), exponential and logarithmic functions like exp(x) and ln(x), roots like sqrt(x), and combinations. Constants pi and e are also supported. Use ^ for exponents and standard function notation.

Reference this content, page, or tool as:

"Simpson's Rule Calculator" at https://MiniWebtool.com/simpson-s-rule-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-05

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Simpson's Rule Calculator

About Simpson's Rule Calculator

Key Features

How to Use the Simpson's Rule Calculator

Simpson's 1/3 Rule Explained

Simpson's 3/8 Rule Explained

Error Comparison

When to Use Each Rule

Supported Functions

Frequently Asked Questions

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Method	Error Order	Error Bound	Exact For
Trapezoidal	\( O(h^2) \)	\( \frac{(b-a)^3}{12n^2} \max\|f''\| \)	Linear functions
Simpson's 1/3	\( O(h^4) \)	\( \frac{(b-a)^5}{180n^4} \max\|f^{(4)}\| \)	Cubics and below
Simpson's 3/8	\( O(h^4) \)	\( \frac{(b-a)^5}{80n^4} \max\|f^{(4)}\| \)	Cubics and below